I/O Efficient Algorithms for the Minimum Cut Problem on Unweighted Undirected Graphs

Abstract

The problem of finding the minimum cut of an undirected unweighted graph is studied on the external memory model. First, a lower bound of Ω((E/V) Sort(V)) on the number of I/Os is shown for the problem, where V is the number of vertices and E is the number of edges. Then the following are presented, for M = Ω(B ²), (1) a minimum cut algorithm that uses $O(c \log E ({\rm MSF}{(V,E)} + \frac{V}{B} {\rm Sort}({V})))$ I/Os; here MSF(V,E) is the number of I/Os needed to compute a minimum spanning tree of the graph, and c is the value of the minimum cut. The algorithm performs better on dense graphs than the algorithm of [7], which requires O(E + c ² V log(V/c)) I/Os, when executed on the external memory model. For a δ-fat graph (for δ > 0, the maximum tree packing of the graph is at least (1 + δ)c/2), our algorithm computes a minimum cut in O(c logE (MSF(V,E) + Sort(E))) I/Os. (2) a randomized algorithm that computes minimum cut with high probability in $O(c \log E \cdot{\rm MSF}{(V,E)} + {\rm Sort}{(E)} \log^2 V + \frac{V}{B} {\rm Sort}{(V)} \log V)$ I/Os. (3) a (2 + ε)-minimum cut algorithm that requires O((E/V) MSF(V,E)) I/Os and performs better on sparse graphs than our exact minimum cut algorithm.